Question

How do you solve the following linear system: `8x+3y-6z=4, x-2y-z=2, 4x+y-2z=-4`?

Answer 1

Combine `2` pairs of equations to eliminate 1 of the variables from 2 equations; then solve the 2 equations for each of the remaining variables.
`color(white)("XXX")(x,y,z)=(-4,0,-6)`

Explanation:

Given:
[1]`color(white)("XXX")8x+3y-6z=4`
[2]`color(white)("XXX")x-2y-z=2`
[3]`color(white)("XXX")4x+y-2z=-4`

To eliminate `x` from 2 equations:

Multiply [2] by `8`
[4]`color(white)("XXX")8x-16y-8z=16`
Subtract [4] from [1]
[5]`color(white)("XXX")19y+2z=-12`

Multiply [3] by `2`
[6]`color(white)("XXX")8x+2y-4z=-8`
Subtract [6] from [1]
[7]`color(white)("XXX")y-2z=12`

Now we have two equations in two unknowns:
[5]`color(white)("XXX")19y+2z=-12`
[7]`color(white)("XXX")y-2z=12`

Adding [5] and [7]
[8]`color(white)("XXX")20y = 0`
[9]`color(white)("XXX")y=0`

Substituting `0` for `y` in [7] (could have used [5])
[10]`color(white)("XXX")(0)-2z=12`
[11]`color(white)("XXX")z=-6`

Substituting `(-6)` for `z` and `0` for `y` in [2] (could have used [1] or [3])
[12]`color(white)("XXX")x-2*(0)-(-6) = 2`
[13]`color(white)("XXX")x=-4`